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Concentration phenomena for critical fractional Schrödinger systems

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  • In this paper we study the existence, multiplicity and concentration behavior of solutions for the following critical fractional Schrödinger system

    $\left\{ \begin{array}{*{35}{l}} \begin{align} & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+V(x)u={{Q}_{u}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{u}}(u,v)\ \ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & {{\varepsilon }^{2s}}{{(-\Delta )}^{s}}u+W(x)v={{Q}_{v}}(u,v)+\frac{1}{2_{s}^{*}}{{K}_{v}}(u,v)\ \ \ \ \text{in }{{\mathbb{R}}^{N}} \\ & u,v>0\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{ in }{{\mathbb{R}}^{N}}, \\ \end{align} & \text{ } & \text{ } & {} \\\end{array} \right.$

    where $\varepsilon>0$ is a parameter, $s∈ (0, 1)$, $N>2s$, $(-Δ)^{s}$ is the fractional Laplacian operator, $V:\mathbb{R}^{N} \to \mathbb{R}$ and $W:\mathbb{R}^{N} \to \mathbb{R}$ are positive Hölder continuous potentials, $Q$ and $K$ are homogeneous $C^{2}$-functions having subcritical and critical growth respectively.

    We relate the number of solutions with the topology of the set where the potentials $V$ and $W$ attain their minimum values. The proofs rely on the Ljusternik-Schnirelmann theory and variational methods.

    Mathematics Subject Classification: Primary: 35A15, 35J50; Secondary: 35R11, 58E05.


    \begin{equation} \\ \end{equation}
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